The next possible symmetry dimensions are \(n^2+5\)  (for general path structures) and \(n^2+4\) 
(for projective structures), see [1]}.
In §REF –REF  we will demonstrate that these structures are variational
by exhibiting Lagrangians (of Kropina type). However they are not metrizable:
for the submaximally symmetric projective structure, called Egorov structure, this follows from [2]};
the submaximally symmetric path structure is not a projective structure hence can not be metrizable
by any pseudo-Riemannian metric. This implies the following result:
